How Are Beam Support Reactions Calculated?

In summary, the problem involves finding the point of the centre of gravity for a system with three different loads applied at different points. By using the equations for summing forces and moments, we can calculate the values for B and C, which represent the forces at the supports of the beam. However, the calculated location for the centroid of area 2 is incorrect and should be adjusted to account for the weight distribution of the triangle load.
  • #1
Guillem_dlc
188
17
Homework Statement
For the given loads, determine the reactions at the beam supports.
Relevant Equations
##\sum F=0, \sum M=0##
Figure:
93DC4582-E535-4AAA-852A-3B4990DB20BB.jpeg


My attempt at a solution:
CFB2307A-F404-4DA9-B71D-9CE894D2AA87.jpeg

We know that ##Q=A_T##
We calculate ##Q##:
$$Q=\dfrac{3\cdot 480}{2}+\dfrac{600\cdot 6}{2}+600\cdot 2=3720\, \textrm{lb}$$
Then we look for the point ##\overline{x}## of the centre of gravity:
$$\overline{x_1}=1\, \textrm{ft},\quad \overline{x_2}=3+\dfrac63=5\, \textrm{ft},\quad \overline{x_3}=3+6+\dfrac22=10\, \textrm{ft}$$
$$\overline{x}=\dfrac{\sum x_iQ_i}{Q}=5,84\, \textrm{ft}$$
$$\sum Fx=\boxed{Bx=0}$$
$$\sum Fy=0=By+C-Q=0\rightarrow \boxed{By=1959,2\, \textrm{lb}}$$
$$\sum M_B=2,84Q-6C=0\rightarrow \boxed{C=1760,8\, \textrm{lb}}$$

Would this not be the case in this one? It's just that the solution tells me the following and I don't get that centre of gravity in area 2:
Official solution:
DBC24C39-E30A-46B8-9ACA-FEC836A6E4E1.jpeg

We have
$$R_I=\dfrac12 (3\, \textrm{ft})(480\, \textrm{lb}/\textrm{ft})=720\, \textrm{lb}$$
$$R_{II}=\dfrac12 (6\, \textrm{ft})(600\, \textrm{lb}/\textrm{ft})=1800\, \textrm{lb}$$
$$R_{III}=(2\, \textrm{ft})(600\, \textrm{lb}/\textrm{ft})=1200\, \textrm{lb}$$
Then
$$\xrightarrow{+}\sum F_x=0:\,\, B_x=0$$
$$\sum M_B=0:\,\, (2\, \textrm{ft})(720\, \textrm{lb})-(4\, \textrm{ft})(1800\, \textrm{lb})+(6\, \textrm{ft})C_y-(7\, \textrm{ft})(1200\, \textrm{lb})=0$$
$$C_y=2360\, \textrm{lb}\qquad \mathbf{C}=2360\, \textrm{lb} \uparrow$$
or
$$\sum F_y=0:\,\, -720\, \textrm{lb}+B_y-1800\, \textrm{lb}+2360\, \textrm{lb}-1200\, \textrm{lb}=0$$
$$B_y=1360\, \textrm{lb}\qquad \mathbf{B}=1360\, \textrm{lb}\uparrow$$
 
Physics news on Phys.org
  • #2
The effective loads are applied at the centroids of the areas. Check ##x_2##
 
Last edited:
  • Like
Likes Lnewqban
  • #3
The location that you have calculated for the centroid of A2 is not correct.
Much of the weight is located towards the right side of the 11-foot beam, therefore the calculated concentrated total weight should be located far from 5.84 feet from the left end.
 
  • #4
Of course, I have taken it as if the triangle is rotated. It should be ##2/3##
 
  • Like
Likes Lnewqban

FAQ: How Are Beam Support Reactions Calculated?

What are reactions at the beam supports?

Reactions at the beam supports refer to the forces and moments that are applied to a beam at its endpoints or supports. These reactions are necessary to keep the beam in equilibrium and prevent it from collapsing.

How do you calculate reactions at the beam supports?

To calculate reactions at the beam supports, you need to apply the principles of statics and equilibrium. This involves summing up all the forces and moments acting on the beam and setting them equal to zero. The resulting equations can then be solved to determine the reactions at the supports.

What factors affect reactions at the beam supports?

The magnitude and direction of the reactions at the beam supports are influenced by several factors, including the type of support (fixed, pinned, or roller), the load applied to the beam, and the geometry of the beam (length, cross-sectional shape, etc.). The location of the supports along the beam also plays a significant role in determining the reactions.

How do you determine the type of support needed for a beam?

The type of support needed for a beam depends on the specific conditions and requirements of the structure. For example, if the beam needs to be fixed in place and prevent rotation, a fixed support would be necessary. If the beam needs to be able to rotate, a pinned or roller support may be used. The location and magnitude of the expected loads should also be considered when determining the type of support needed.

Can reactions at the beam supports change over time?

Yes, reactions at the beam supports can change over time due to various factors such as temperature changes, structural settlement, or external forces. It is essential to consider the potential for these changes when designing and analyzing a structure to ensure its stability and safety.

Similar threads

Replies
13
Views
1K
Replies
7
Views
6K
Replies
6
Views
7K
3
Replies
104
Views
15K
2
Replies
52
Views
10K
Back
Top